I feel very on-board with this research aesthetic.
Here are just some nit-picks/notational confusions I had while reading this;
The sequence p,step(p),step(step(p)),…, i.e., n↦stepn(p), is the computation seeded at p (or a “trajectory” in dynamical systems terminology).
...
A property P is achieved by a computation s if there exists some number of steps n such that s(n)∈P…
It took me a second to figure out what s(n) referred to, partly because the first s was not rendered in LaTeX, partly because it was never shown as a function before, and partly because it looked kinda like stepn(p), so I thought maybe the notation had changed.
the empty board C={⊥}
I’ve seen ⊥ as “false” before, but I don’t think it’s super common, and you also previously said
a pattern is an infinite two-dimensional Boolean grid, or equivalently a function of type ℤxℤ→{true, false}
which made this feel like a switchup of notation. (Also, I think the type signature is off? The empty board C should be a function, but instead it’s a set containing one symbol...)
This includes still lifes (N=0), blinkers (N=2)
I think if blinkers have period 2 then still lifes have to be considered to have period 1, and not 0.
Eater. An eater p is robust for P={p} within any context c that contains n≥0 spaceships traveling in the direction of the eater (and nothing else on the board).
I think the true thing is a lot weaker than this; it’s robust to gliders (not all spaceships) traveling along a specific diagonal with respect to the location of the eater (and possibly the glider has to have a certain phase, I’d have to check).
The basin of attraction for a pattern p and a property P is the largest context set B such that p is robust for P within B.
Examples:
Eater. Let p be an eater and P={p}. B is the context set containing n≥0 spaceships moving in the direction of the eater and nothing else (in any other context, the contents of the board don’t get consumed by the eater).
This is definitely not the largest context set B, because there are tons of patterns that extinguish themselves.
I feel very on-board with this research aesthetic.
Here are just some nit-picks/notational confusions I had while reading this;
It took me a second to figure out what s(n) referred to, partly because the first s was not rendered in LaTeX, partly because it was never shown as a function before, and partly because it looked kinda like stepn(p), so I thought maybe the notation had changed.
I’ve seen ⊥ as “false” before, but I don’t think it’s super common, and you also previously said
which made this feel like a switchup of notation. (Also, I think the type signature is off? The empty board C should be a function, but instead it’s a set containing one symbol...)
I think if blinkers have period 2 then still lifes have to be considered to have period 1, and not 0.
I think the true thing is a lot weaker than this; it’s robust to gliders (not all spaceships) traveling along a specific diagonal with respect to the location of the eater (and possibly the glider has to have a certain phase, I’d have to check).
This is definitely not the largest context set B, because there are tons of patterns that extinguish themselves.
Thanks Alex for the detailed feedback! I have updated the post to fix these errors.
Curious if you have high-level thoughts about the post and whether these definitions have been useful in your work.